| L(s) = 1 | + 2·2-s + 3·4-s + 3·5-s − 4·7-s + 4·8-s − 3·9-s + 6·10-s + 3·11-s − 5·13-s − 8·14-s + 5·16-s − 3·17-s − 6·18-s − 5·19-s + 9·20-s + 6·22-s + 3·23-s + 5·25-s − 10·26-s − 12·28-s + 3·29-s − 8·31-s + 6·32-s − 6·34-s − 12·35-s − 9·36-s + 7·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.34·5-s − 1.51·7-s + 1.41·8-s − 9-s + 1.89·10-s + 0.904·11-s − 1.38·13-s − 2.13·14-s + 5/4·16-s − 0.727·17-s − 1.41·18-s − 1.14·19-s + 2.01·20-s + 1.27·22-s + 0.625·23-s + 25-s − 1.96·26-s − 2.26·28-s + 0.557·29-s − 1.43·31-s + 1.06·32-s − 1.02·34-s − 2.02·35-s − 3/2·36-s + 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.392123169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.392123169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.55531997267813121249278314769, −13.05462767031080919634940370834, −12.81902541248718095740813569189, −12.43664734168969743992896652938, −11.56789158902987700790310893324, −11.40599073417725308707227547191, −10.46822610720075562760876060459, −10.15311706372367480395942703027, −9.438248087041028602410420494484, −9.125235938597250740275394424889, −8.407100253915533330320315584223, −7.25688022136843343210620374945, −6.73568279451834012808834900322, −6.38984903996640421068437877837, −5.79569964434831973108258461910, −5.29691684266655132155212203925, −4.45103574423999985661409220879, −3.63371565836137938835886188259, −2.69731698787138918160492405238, −2.25602393735114106600285449273,
2.25602393735114106600285449273, 2.69731698787138918160492405238, 3.63371565836137938835886188259, 4.45103574423999985661409220879, 5.29691684266655132155212203925, 5.79569964434831973108258461910, 6.38984903996640421068437877837, 6.73568279451834012808834900322, 7.25688022136843343210620374945, 8.407100253915533330320315584223, 9.125235938597250740275394424889, 9.438248087041028602410420494484, 10.15311706372367480395942703027, 10.46822610720075562760876060459, 11.40599073417725308707227547191, 11.56789158902987700790310893324, 12.43664734168969743992896652938, 12.81902541248718095740813569189, 13.05462767031080919634940370834, 13.55531997267813121249278314769
